The images on this page are created using the standard iterative
series of the Mandelbrot, that is, iterate the function
zn+1 = zn2 + z0
where z0 is each point in the image plane (complex plane).
However, instead of recording the behavior of the series at each
point z0 we now consider only those points that escape
to infinity and we create a density plot of the terms in the
series. The result then is a 2D density plot of the trajectories that
escape to infinity. The following shows the buddhabrot for that part
of the complex plane that is interesting.

Complex plane: -13/6 <= real <= 7/6 and
-5/3 <= imaginary <= 5/3

In the following, the boundary of the Mandelbrot
is shown in the same coordinate system in yellow.
Note that images here have the imaginary axis horizontal and the
real axis vertically, this is a 90 degree clockwise rotation from the
way the Mandelbrot is traditionally drawn.

One unfortunate consequence of how these images are created is that
zooming in becomes increasingly inefficient. Points on an escape
trajectory that pass through an arbitrary small region of the
complex plane could have come from distant regions. So when computing
the buddhabrot of a zoomed in portion of the complex plane, one
still needs to choose z0 values from the whole plane.
This isn't quite true, since all points outside the circle of
radius 2 escape monotonically one only has to sample points to the region
around the origin. As the area being computed gets smaller the probability of
a trajectory passing through it deceases so it takes more iterations
before the density plot becomes significant.

While the normal Mandelbrot is "dull" on the interior, the buddhabrot
has lots of internal structure.

Unlike the Mandelbrot where choosing the number of iterations that are
tested in order to determine whether the point escapes or not (often called
the "bailout") doesn't have a profound effect
on the actual structure, in the buddhabrot it can. For example in the following
the first uses a bailout of 200 and the second uses 20. For very high quality
versions of the buddhabrot a bailout above 1000 is suggested, of course that
increases compute times.

Bail-out: 200

Bail-out: 20

There are a number of ways one might colour the buddhabrot, the most common is
to compute the buddhabrot a number of times with different number of interations.
Each of these images is mapped to a colour and combined. One can similarly combine
images with different bailout values (right), a common technique in astrophotography
for combining images of different "exposures".